These questions below are culled from old quizzes and exams and represent the
level of difficulty that youll face on Fridays exam. Note that the exam will have
both conceptual as well as computational questions, so be sure to be able to answer
the types o
Math 126: Exam 3 Review
November 2016
This list of questions is representative, but not exhaustive, of what you will see
on Fridays Exam. Use these as a guide to help you study. We will go over solutions
to these on Wednesday in class.
1. Find the followi
1. Find the first five terms of the Maclaurin Series for f (x) =
x = 0.
1 x centered at
2. (a) Find a series representation for f (x) = ex .
(b) Find a series representation for g(x) = xe3x
(c) Find g (7) (0), that is, the seventh derivative of g evaluate
Math 126: Ancillary Review Problems
These problems come largely from the Final Exam in Fall 2014. I will post a key for
these by Sunday.
1. Determine whether the following series converge or diverge. Give an appropriate justification in each case using ou
Math 126: Final Exam Review
Fall 2016
These problems come largely from the Fall 2015 Final Exam. We will go over these
in the review session on Sunday evening.
1. Determine whether the following series converge or diverge. Give an appropriate justificatio
Selected Solutions: Reading HW, Chapter 1
2. A mathematical model is a DE that describes some physical process.
4. An equilibrium solution is a solution that never changes in time (a constant solution).
6. The trade o in modeling: (p. 15) Accuracy versus
Quick Overview: Complex Numbers
February 24, 2012
1
Initial Denitions
Denition 1 The complex number z is dened as:
z = a + bi
where a, b are real numbers and i =
(1)
1.
Remarks about the denition:
Engineers typically use j instead of i.
Examples of comp
Solutions: Section 2.1
1. Problem 1: See the Maple worksheet to get the direction eld. You should see that it
looks like all solutions are approaching some curve (maybe a line?) as t . To be
more analytic, let us solve the DE using the Method of Integrati
Summary: Chapter 6, 5.1-5.3
Here is a summary of the material in Chapter 6 and sections 5.1-5.3.
6.1
Know the denition of the Laplace transform.
When will the Laplace transform exist? If the function is piecewise continuous and
of exponential order. Kno
Exam 2 Summary
Notes
The exam will cover material from Section 3.1 to 3.8. We did not get to the last part of 3.8
(damping and forcing), so it will not be on the exam (but the part on beating and resonance
may be).
There are two sets of formulas that will
Solutions to the Complex Number Exercises
1. Suppose the roots to a cubic polynomial are a = 3, b = 12i and c = 1+2i. Compute (xa)(xb)(xc).
SOLUTION: You could multiply it all out at once. If you take (x b)(x c) rst, you get:
(x a)(x b)(x c) = (x 3)(x2 2x
Study Guide: Exam 1, Math 244
The exam covers material from Chapters 1 and 2 (up to 2.6), and will be 50 minutes in length. You may
not use the text, notes, colleagues or a calculator.
Because a dierential equation denes a function (the solution), there a
Solutions: Section 2.2
2.2, 1: Give the general solution: y = x2 /y
y dy = x2 dx
1 2 1 3
y = x +C
2
3
2.2, 3: Give the general solution to y + y 2 sin(x) = 0.
First write in standard form:
dy
= y 2 sin(x)
dx
1
dy = sin(x) dx
y2
Before going any further,
Solutions: Section 2.3
2.3, 3: In this model of salt in a tank of water, let Q(t) be the amount of salt (in
pounds) at time t (measured in minutes). Then dQ/dt will be measured in pounds per
minute.
The rate in, at least initially:
1 lbs
gal
2
= 1 lbs/mi
Selected Solutions, Section 6.5
2. Solve y + 4y = (t ) (t 2), y(0) = 0 and y (0) = 0.
Its useful to think about the problem before solving it:
Up until time t = , we have a spring-mass system with no damping and
no initial displacement or velocity. Left a
Solutions: Section 2.4
1. Problems 1-6 ask you to apply the Existence and Uniqueness Theorem to a given linear
ODE. Be sure to put the DE in standard form rst! Some notes as you do this:
The interval is a single (connected) interval.
For theoretical rea
Selected Solutions, Section 6.6
2. You can choose almost any function, even 1 1:
t
t
11=
x =t
1 dx =
0
0
3. The following trig identity is used1
sin(A) sin(B) =
1
[cos(A B) cos(A + B)]
2
Then:
t
sin(t) sin(t) =
sin(t x) sin(x) dx =
0
1
2
t
cos(t 2x) cos(t
Help on section 6.2
The main ideas:
Take the Laplace transform of a DE.
Solve it for Y (s) (the transform of y(t).
Invert the transform- This may involve partial fractions and/or completing the square.
The rst set of exercises focuses on this last part
Selected Solutions: Section 6.1
1. This is piecewise continuous, but not continuous at t = 1.
2. Not continuous and not piecewise continuous.
3. Continuous (so also piecewise continuous).
5.
(a) Find the Laplace transform of t (done in class).
(b) Find th
Section 6.3 Homework Notes
2. To sketch the graph, try rst rewriting what is given as a piecewise dened function.
The function:
(t 3)u2 (t) (t 2)u3 (t)
depends on the value of t:
If t < 2, the function is zero, since u2 and u3 are zero.
If 2 t < 3, then
Review Questions: Exam 3
1. What is the ansatz we use for y in: Chapter 6? Section 5.2?
2. Finish the denitions:
The Heaviside function, uc (t):
The Dirac function: (t c) (Note: the Dirac function should be dened as a
certain limit)
Dene the convolutio
Solutions to the Review Questions
Short Answer/True or False
1. True or False, and explain:
(a) If y = y + 2t, then 0 = y + 2t is an equilibrium solution.
False: This is an isocline associated with a slope of zero, and furthermore, y = 2t is not a solutio
Sample Question Solutions (Chapter 3, Math 244)
1. State the Existence and Uniqueness theorem for linear, second order dierential equations (non-homogeneous is the most general form):
SOLUTION:
Let y + p(t)y + q(t)y = g(t), with y(t0 ) = y0 and y (t0 ) =
Solutions to Review Questions: Exam 3
1. What is the ansatz we use for y in
Chapter 6? SOLUTION: y(t) is piecewise continuous and is of exponential order
(so that Y (s) exists).
Section 5.2? SOLUTION: y(x) is analytic at x = x0 . That is,
an (x x0 )n
y(
Review Questions, Exam 1, Math 244
These questions are presented to give you an idea of the variety and style of question that
will be on the exam. It is not meant to be exhaustive, so be sure that you understand the
homework problems and quizzes.
Discuss
Sample Questions (Chapter 3, Math 244)
Exam Notes: You will not be allowed to have a calculator or any notes. You will have the
formulas listed in the Formula Page on the summary.
1. State the Existence and Uniqueness theorem for linear, second order dier
Final Exam Review: Math 244
Instructions at the top of the exam will be the following:
Instructions:
A table of Laplace transforms and the Poicar Diagram will be provided on a separate
e
sheet. The Variation of Parameters formula will be provided on the b
Homework Set 1 Solutions
1. Exercise 22, p 363 (Section 7.1)
Construct the DEs by looking at Rate In - Rate Out, and make sure
your units are matching up. Dene Q1 (t) and Q2 (t) to be the ounces of
salt in Tanks 1 and 2 (respectively). Before we start, we